Leave $ G $ be a group of complex semisimple lies, and let $ H $ be a subgroup corresponding to a subset of the extended Dynkin diagram of $ G $ (to the Borel – of Siebenthal). I would like to know if there is a recipe to calculate the normalizer of $ H $. My feeling is that this must be known, but I could not find anything.
To be specific, consider this example. Leave $ G = E_7 $ (simply connected). There is a subgroup of type $ A_7 $, which has an index $ 2 $ Inside your normalizer. This corresponds to the involution of the Dynkin type diagram. $ A_7 $, which one can check respects the highest root of $ E_7 $ And so it respects the extended Dynkin type diagram. $ E_7 $. On the other hand, consider the type subgroup $ D_6 times A_1 $. The index of the subgroup in its normalizer is at most $ 2 $, because the Dynkin diagram has a group of automorphism $ 2 $. It is clear that the obvious involution of the Dynkin diagram does not extend to an involution of the extended Dynkin diagram $ E_7 $, because the highest root is sent to some other root. Does this imply that the subgroup is self-normalizing? I hope that this subgroup does not self-normalize.